We present and discuss a framework for computer-aided multiscale
analysis, which enables models at a fine (microscopic/
stochastic) level of description to perform modeling tasks at a
coarse (macroscopic/systems) level. These macroscopic modeling
tasks, yeilding infomration over long time and large
scales, are accomplished through approximately initialized calls to
the microscopic simular for only short times and small
spatial domains. Traditionally modeling approaches first involve the
derivation of macroscopic evolution equations (balances closed through
constitutive realtions). An arsenal of analytical and numerical
techniques for the efficent solution of such evolution equations
(usually Partial Differential Equations, PDEs) is then brought to bear
on the problem. Our equation-free (EF) approach, introduced in [1],
when successful, can bypass the derivation of the macroscopic evolution
equations when these equations conceptually exist but are not
available in closed form. We discuss how the mathmatics-assisted
development of a computational superstructure may enable alternative
descriptions of the problem physics (e.g. Lattice Boltzmann (LB),
kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic
simulators, executed over relatively short time and space scales) to
perform systems level tasks (integration over relatively large time and
space scales, "coarse" bifurcation analysis, optimization, and control)
directly. In effect, the procedure constitutes a system identification
based, "closure-on-demand" computational toolkit, bridging microscopic/
stochastic simulation with traditional continuum scientific computation
amd numerical analysis. We will breifly survey the application of these
"numerical enabling technology" ideas through examples including the
computation of coarsely self-similar solutions, and discuss various
features, limitations and potential extensions of the approach.